I'm trying to perform search by tag using the request like this:
http://overpass.osm.rambler.ru/cgi/interpreter?data=[out:json];node[%22addr:postcode%22=20156];out;
Is it possible to query only for nodes that has been added/modified after some arbitrary date?
Yes, there are the two keywords changed and newer. Better use newer since changed requires lots of memory to run, see issue 278.